Zermatt or Chamonix

As far as activities for your toddler, both are quite comparable.
Zermatt has Wolli Park that is fun in the summer time. Besides that there is tons of hiking that would be appropriate for a very small child, lots of lifts and gondolas to ride, walking inside a glacier and going down an ice slide up at Glacier Paradise, and the Valais Blacknose Sheep who live in the mountains in the surrounding area are about the cutest things ever - your child would love them! The main part of Zermatt really doesn't have a lot for kids. It's mainly a touristy shopping area with hotels scattered throughout.
Chamonix also has lots of hiking of various types all easily reached by gondola. The town is also mainly a touristy shopping district. There is a small amusement park in Chamonix that has things a toddler would enjoy doing with you. The cogwheel train that goes up to the Mer du Glace is fun as well and you can go inside the glacier although since it has melted away so dramatically you now have to walk down hundreds of steel stairs to get to it.
For me, it would come down to the quality of the Christmas card shots I could get (Zermatt!) or the MUCH lower cost option (Chamonix!). You'll be happy no matter which you choose...they are equally beautiful and dramatic, will have similar weather, and you'll have a great time either way.

My vote is for Chamonix, and if the weather is good be sure to take the cable cars up and over to Italy - spectacular views of Mt Blanc.

While I agree that you will enjoy either destination, I also vote for Chamonix as more affordable (and in my memory, somehow more friendly). If you are driving, Chamonix has this animal park. Either town is accessible by public transport, but usually with connections. If you're coming into Geneva airport, then the most direct route to Chamonix is by airport bus or taxi (www.chamonix-transfer.com). If you're not in a hurry, the train ride around the lake is scenic (3.5 hours with 2 connections).

We finally went to Zermatt and had lovely time. We stayed at hotel opposite McDonalds on Bahnofstrasse. It was very convenient.
My little boy enjoyed Wolli and enjoyed our hikes and trips to Matterhorn. In the evenings we did walks on river side and in market area. It was perfect place to go with a toddler.

So glad you have a great time!